Example 29.43.2 (07ZS)—The Stacks project

Example 29.43.2. Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_ S$-algebra generated by $\mathcal{A}_1$ over $\mathcal{A}_0$. Assume furthermore that $\mathcal{A}_1$ is of finite type over $\mathcal{O}_ S$. Set $X = \underline{\text{Proj}}_ S(\mathcal{A})$. In this case $X \to S$ is projective. Namely, the morphism associated to the graded $\mathcal{O}_ S$-algebra map

\[ \text{Sym}_{\mathcal{O}_ X}^*(\mathcal{A}_1) \longrightarrow \mathcal{A} \]

is a closed immersion, see Constructions, Lemma 27.18.5.

Comments (2)

Comment #8109 by Laurent Moret-Bailly on February 16, 2023 at 05:33

This is repeated later as Lemma 31.30.5. But the condition on $\mathcal{A}$ can be relaxed, at least if $S$ is quasicompact: by 10.56.2 and 27.11.8 it is enough to assume that $\mathcal{A}$ is finitely generated over $\mathcal{A}_0$ and $\mathcal{A}_0$ is finite over $\mathcal{O}_S$ . This is useful in practice, but I could not find this statement in SP (or in EGA!).

Comment #8220 by Stacks Project on March 03, 2023 at 12:51

Going to leave this alone. Of course, I agree with you.

Comments (2)

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